For many, mathematics is a stultifying series of rules and mechanical computation procedures guided by, at worst, nothing but rote memorisation and at best, intuitions and heuristics. Outside of the very best advanced classes the concept of mathematical argumentation is foreign and the idea of mathematics being creative or beautiful is downright alien.
But formal mathematics could not be more different from the perception of mathematics fostered by schools. Formal mathematics is unlike almost every other domain of human inquiry, in that it entails the almost pure application of logic. The fundamental analytic tool of mathematics is the mathematical proof, a sequence of logical steps from one conclusion to the next, originating in axioms or statements that are assumed to be true. From such axioms one can derive stable, truly complete and artful conclusions about the world and the nature of reality.
The creative potential inherent to mathematics can be seen when examining the key process of abstraction. Abstraction is the method of collecting the ‘essence’ of a phenomena of interest and generalising to capture a wider set of phenomena. The process and power of abstraction can be seen in the very earliest mathematical developments of ancient civilisations. Many ancient cultures developed basic mathematical computational tools to measure distances, areas and volumes. The ancient Egyptians were particularly interested in consistent measurement tools for the proper placement of things like property markers, which tended to wash away in the Nile’s yearly floods.
One technique used by the Egyptians was to take a loop of rope and create marks at 12 equal intervals. The Egyptians knew if they then took this loop of rope and formed a triangle of sides 3, 4 and 5 intervals each, a right angle (90 degrees) would be formed at the corner of the 3 interval and 4 interval sides. This fact has obvious utility in marking out squares or rectangles (perhaps for fields), which have 4 corners at right angles.
To the Egyptians this was a procedure with straight forward practical use. Nowadays any high school student can recognise this as a simple application of the Pythagorean theorem. The theorem is named for Pythagoras (570 BC – 495 BC), the Greek mathematician and cultist who supposedly provided the first proof. The theorem states that for any right-angle triangle, the sum of the square of the two shortest sides is equal to the square of the longest side.
Modern high school students can recognise this technique and its broader significance thanks to the mathematical and philosophical advancements made by the Greeks, who took something like the Egyptian procedure for forming a specific right-angle triangle and generalised it to a truth about the dimensions of any triangle. As trivial as this sounds to modern ears, the use of the word ‘any’ requires a large philosophical and conceptual leap.
One can imagine that Pythagoras’ predecessors would have confirmed many specific cases of his eponymous theorem (possibly through practical work like the Egyptian measuring implement). With the success they had, perhaps over time a kind of ‘scientific’ inductive belief grew to hold sway over the minds of his predecessors – a belief that the theorem held for any right triangle. The ‘scientific’ version of the theorem could be and probably was used for practical purposes. However, the key thing that distinguishes mathematical truth from scientific truth is the fact that mathematical truth holds deductively so long as we accept its axioms. To establish the theorem as mathematical truth, Pythagoras and the Greeks saw that they needed to make use of abstraction. The translation from concrete examples and procedures to the abstract requires taking the essence of the specific case and generalising to understand a broader phenomenon. The Greeks re-conceptualized the triangle as no longer a specific shape drawn in sand or etched in clay. They removed the unimportant details and re-framed it as a concept to capture any closed shape with three straight line segment sides and three vertices.
This is a bigger leap than it appears at first. Before the Greeks, the triangle and more generally geometric shapes were defined by their representation as etchings on clay or lines in the sand. After the Greeks, representations came to be defined by their abstract conceptualisation. We were able to draw an imperfect representation of a triangle on a piece of clay yet not lose sight of the fact that we were symbolically representing a broader idealised concept.
The power of this abstraction can be best shown in the simple diagram proof of Pythagoras’ theorem:
The proof proceeds by first noting that both outer squares have the same area due to having the same side length (a + b). Similarly, the triangles in each square have the same area (we will call this area TriangleArea) because they have the same dimensions. This leads us to the conclusion that, by the equivalent areas of the outer squares the sum of the internal shape areas must also be equal:
We can subtract the area of the 4 triangles from both sides and still retain our equality (we are subtracting the same amount from two equal things) to obtain:
This is the desired result, QED.
It should be clear from the above that even with this diagram we are not taking a specific triangle or square for this proof. The triangles and squares are simply representations or visual aids for any given right triangle. The actual properties of the triangles and squares utilized are from our abstract definition. While it would be much more difficult to illustrate, the entire argument could have been made in words with the abstract definitions, highlighting the unimportance of the diagram to the truth of the statement. The power of abstraction is clear. Once we re-frame our definitions to make sense outside of tangible representations, we can simply prove the Pythagorean theorem for any triangle in essentially two lines of actual mathematical statements.
The beauty of mathematics is that it is not static. Such definitions often generate new questions. For example, we could now ask of our triangle definition: how do we precisely define abstract lines at all and what makes a line straight? The famous Euclid would seek to answer these new questions in his 4th century BC book, Elements, a text filled with hundreds of precise mathematical proofs.
It is easy to see how abstraction requires creative thinking. The process is not simply an exercise in computational logic – it requires an ability to think imaginatively and inventively. It is not a machine-like process that takes us from our tangible definition of the triangle to our abstract definition but a process that requires the artfulness of the poet or the painter. Perhaps if more students were exposed to the underlying creative process inherent to mathematics, a subject dreaded by many could be made to reflect the liveliness of the reality it so perfectly describes.